# American Institute of Mathematical Sciences

• Previous Article
Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions
• DCDS Home
• This Issue
• Next Article
From gradient theory of phase transition to a generalized minimal interface problem with a contact energy
May  2016, 36(5): 2757-2779. doi: 10.3934/dcds.2016.36.2757

## Random attractor of stochastic Brusselator system with multiplicative noise

 1 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, United States

Received  December 2014 Revised  August 2015 Published  October 2015

Asymptotic dynamics of stochastic Brusselator system with multiplicative noise is investigated in this work. The existence of random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimations than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the attracting regularity to be an $(L^2\times L^2,H^1\times H^1)$ random attractor.
Citation: Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757
##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] A. V. Babin and M. I. Vishik, Regular attractors of semigroups of evolutionary equations,, J. Math. Pures Appl., 62 (1983), 441.   Google Scholar [3] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differential Equations, 246 (2009), 845.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [4] T. Caraballo, J. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation,, Proc. R. Soc. Lond. A, 457 (2001), 2041.  doi: 10.1098/rspa.2001.0819.  Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002).   Google Scholar [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [7] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [8] F. Flandoli and B. Schmalfu$\beta$, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21.  doi: 10.1080/17442509608834083.  Google Scholar [9] M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Modles in Biology, Chemistry and Population Genetics,, Springer, (2012).  doi: 10.1007/978-3-642-22664-9.  Google Scholar [10] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London, 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar [11] L. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems,, J. Chem. Physics, 48 (1968), 1695.   Google Scholar [12] J. C. Robinson, Stability of random attractors under perturbation and approximation,, Journal of Differential Equations, 186 (2002), 652.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar [13] J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge Univ. Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar [14] R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar [15] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 269.  doi: 10.3934/dcds.2014.34.269.  Google Scholar [16] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonl. Anal. A, 75 (2012), 3049.  doi: 10.1016/j.na.2011.12.002.  Google Scholar [17] Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 301.  doi: 10.3934/dcds.2014.34.301.  Google Scholar [18] W. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise,, Nonl. Anal. A, 84 (2013), 61.  doi: 10.1016/j.na.2013.01.014.  Google Scholar

show all references

##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] A. V. Babin and M. I. Vishik, Regular attractors of semigroups of evolutionary equations,, J. Math. Pures Appl., 62 (1983), 441.   Google Scholar [3] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differential Equations, 246 (2009), 845.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [4] T. Caraballo, J. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation,, Proc. R. Soc. Lond. A, 457 (2001), 2041.  doi: 10.1098/rspa.2001.0819.  Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002).   Google Scholar [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [7] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [8] F. Flandoli and B. Schmalfu$\beta$, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21.  doi: 10.1080/17442509608834083.  Google Scholar [9] M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Modles in Biology, Chemistry and Population Genetics,, Springer, (2012).  doi: 10.1007/978-3-642-22664-9.  Google Scholar [10] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London, 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar [11] L. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems,, J. Chem. Physics, 48 (1968), 1695.   Google Scholar [12] J. C. Robinson, Stability of random attractors under perturbation and approximation,, Journal of Differential Equations, 186 (2002), 652.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar [13] J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge Univ. Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar [14] R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar [15] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 269.  doi: 10.3934/dcds.2014.34.269.  Google Scholar [16] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonl. Anal. A, 75 (2012), 3049.  doi: 10.1016/j.na.2011.12.002.  Google Scholar [17] Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 301.  doi: 10.3934/dcds.2014.34.301.  Google Scholar [18] W. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise,, Nonl. Anal. A, 84 (2013), 61.  doi: 10.1016/j.na.2013.01.014.  Google Scholar
 [1] Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020060 [2] Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 [3] Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887 [4] Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020055 [5] Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269 [6] Tomás Caraballo, I. D. Chueshov, Pedro Marín-Rubio, José Real. Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 253-270. doi: 10.3934/dcds.2007.18.253 [7] Julia García-Luengo, Pedro Marín-Rubio. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2127-2146. doi: 10.3934/cpaa.2020094 [8] Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120 [9] Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1 [10] Junwei Feng, Hui Liu, Jie Xin. Uniform attractors of stochastic two-compartment Gray-Scott system with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020116 [11] T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz. Dynamics of some stochastic chemostat models with multiplicative noise. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1893-1914. doi: 10.3934/cpaa.2017092 [12] Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441 [13] Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058 [14] Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875 [15] Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020028 [16] Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054 [17] Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081 [18] Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087 [19] Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047 [20] T. Tachim Medjo. The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 177-197. doi: 10.3934/dcdsb.2010.14.177

2018 Impact Factor: 1.143